Modified Baryonic Dynamics: Two-Component Cosmological Simulations with Light Sterile Neutrinos

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Modified Baryonic Dynamics: two-component cosmological simulations with light sterile neutrinos

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Please note that terms and conditions apply. JCAP10(2014)079 hysics P and le a,e ic t ar doi:10.1088/1475-7516/2014/10/079 G. Gentile d strop A B. Famaey, b,c, f osmology and and osmology C A. Diaferio, a

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ou An IOP and SISSA journal An IOP and 2014 IOP Publishing Ltd and Sissa Medialab srl Astrophysics, Cosmology & Gravity Centre,Private Dept. Bag of X3, Astronomy, University Rondebosch, of 7701 Cape South Town, E-mail: Africa , [email protected] [email protected], [email protected], , [email protected] [email protected] Department of Physics andPleinlaan Astrophysics, 2, Vrije Universiteit Brussels, Brussel, 1050 Belgium Dipartimento di Fisica, Universitàdi Torino, Via P. Giuria 1, Torino,Istituto I-10125 Nazionale Italy di FisicaVia Nucleare, P. Giuria 1, Torino,Observatoire I-10125 astronomique Italy de Strasbourg,11 CNRS rue UMR de 7550, l’Université,Strasbourg, Universitéde F-67000 Strasbourg, Sterrenkundig France Observatorium, Universiteit Gent, Krijgslaan 281, Gent, 9000 Belgium b c e d a f c

Received July 7, 2014 Revised September 3, 2014 Accepted October 8, 2014 Published October 30, 2014 K.J. van der Heyden G.W. Angus, Modified Baryonic Dynamics: two-component cosmological simulations with light sterile neutrinos J JCAP10(2014)079 1407.1207 modified gravity, cosmological simulations, dark matter simulations, galaxy In this article we continue to test cosmological models centred on Modified New- Abstract. tonian Dynamics (MOND)to with solve light the sterile fine-tuningsuccessful neutrinos, problems predictions which of on could the largermological in standard model, scales. principle model here be Due we onproduced a to test galaxy only way a previous scales by speculative failures while(hence the model of preserving Modified baryons where the Baryonic and the simple Dynamics).separate the modified MOND gravitational the sterile cos- We field baryonic neutrinos use is attenuate N-body produce two-component the a cosmological particles over-production simulations purely from oforiginal to Newtonian the massive MOND field sterile galaxy plus neutrino cluster lightlation ones. sterile halos notwithstanding, neutrinos the which The Modified scenario. wereamplitude Baryonic premise prevalent Theoretical for Dynamics in is issues model the the to with failspower galaxy to such spectrum cluster produce a normalisation. mass the formu- function correct for any reasonableKeywords: value of theclusters primordial ArXiv ePrint: JCAP10(2014)079 6 – 1 – model would be cosmological N-body simulations capable (or concordance) model to successfully match the acoustic CDM CDM Λ Λ ), with baryons making up a mere 5% of the energy budget. Invoking this CDM 2.2.1 Initial2.2.2 conditions and simulation Two-component setup5 5 simulations Λ Approaching the missing mass problem from the other direction, cosmologists have 2.1 One-component2.24 simulations Modified Baryonic Dynamics4 3.1 Newtonian3.2 equivalent mass6 Renormalising3.3 the mass function7 The significance of the CMB quadrupole8 found a satisfactory(i.e. representation General of Relativity). the Thisenergy comes Universe ( through at the large combinationcombination of scales allows cold using the dark matter standard and gravity dark 1 Introduction Modified Newtonian Dynamics (MOND; [1],to and gravity in see the [2] ultra forics weak-field a of regime. recent galaxies review) It of has is allOn a a types, remarkable modification these shapes ability to and scales, reproduce sizes theregularities the ([3–6]) standard dynam- — in with cosmological galaxy only model properties alarge few and still scales, notable has struggles like caveats ([7,8]). a to clustersa number of reproduce missing of galaxies, the mass other the observed describe problem problems MOND cosmological (see for approach phenomena e.g. does MOND like [9, not(CMB; in the [15–18],10]). work anisotropies clusters but [11–14]. in On of see the There [19]). galaxies. cosmic is microwave There background is also no clear way to peaks in the CMB [20].23] Furthermore, is the observed well distributionthat reproduced of would matter by on confirm large N-body the scales simulations [ 21– of structure formation [24–27]. Something 3 Results of producing galaxiestherein). that replicate These the observed31], properties properties the of of relative galaxies observed frequencies includeand galaxies of — the ([9] the perhaps galaxy most and various luminosity importantly Hubble references Two — function types of dynamical [28– [32], properties the the elegantly most encapsulatedrelation colour significant by bimodality [36, MOND. of [33–35], ]37 thesethe dynamical and inferred properties (ii) are gravitational the (i) fieldincludes scaling the the (from requirement baryonic between a for Tully-Fisher the dark dynamicalthan matter cusped observed measure (DM) [39, density — to40]. have profile [38]). a of centrally This cored baryons distribution, second and rather property Contents 11 Introduction 2 Background and method3 4 Conclusion 10 JCAP10(2014)079 200. So if MOND is the correct ∼ z ) and the phase-space constraints from 3 10 z > – 2 – model gives the correct cluster scale halo mass function at CDM Λ At minimum, the There is another factor to consider here which rules out the idea of MOND not being Using a purpose built MOND cosmological N-body code, [52] (hereafter AD11) showed In summary it does not appear to be possible to form the correct halo mass function Although there are promising efforts to satisfy the aforementioned criteria], [40–45 these Another approach is to assume MOND describes the dynamics of galaxies well and that = 0, whether some additional boost to gravity is required to form the clusters early enough activated until some low redshift. That is that galaxies in MOND must form without the description of gravitational dynamicsnot on as galaxy described scales, above and then(a yet either conspire highly the to contrived produce initial scenario), theis conditions or correct are important CMB perhaps angular because MOND power spectrumCDM, although does the not galaxies clusters affect require clearly the MONDNewtonian do sterile gravity to not neutrinos. reproduces form require the This (and MOND cluster stably mass at function. all exist) and without z one should nothas ignore been how well discussed incluster mass the function literature if ([54–61]).rapid it growth MOND influences and has thegravity, the a sterile but formation double neutrinos. these of negative moremasses Not effect much massive only are larger on does halos further the and it nowsuggest enhanced, denser that facilitate have if causing structures the more MOND poorer MOND than gravitational gravityonly field agreement in is meaning a with not Newtonian their produced Newtonian the by dynamical baryons, the gravitational data. sterile that neutrinos field it (meaning This will ishow result have produced significant a might this by positive influence influence them), is. on but the is halo mass only function. produced Below by we the address this model was inconsistent withhigh the mass observed clusters cluster and massthis function too — behaviour few it low for produced mass MOND toonot clusters. cosmological many limited [53] simulations to (hereafter to 11 A13) eVregardless produce further sterile of too demonstrated neutrinos. many (i) In superclustersMOND the fact, was is all value “switched masses of on”, ofpolating the (iii) sterile function. the neutrinos MOND Other normalisation were factors accelerationbe ruled of like constant, ineffective. out, the the equation (ii) initial of conditions the state and redshift of (iv) dark at the energyin which inter- were standard also shown MOND to fromHarrison-Zel’dovich any power sterile spectrum neutrino under initial GR conditions until that grew from an initially clusters of galaxies [14].that The the added 11 attraction eV ofbe neutrinos this occupied would idea (just (which be likeThe was the fully reviewed other active thermalised, by neutrinos) reason [51]) i.e. —properties was MOND removing half of any plus of neutrinos fine sterile on all tuningwould small neutrinos of quantum leave scales their the is states means abundance. impressive so would they results attractive would of not is MOND influence in that galaxies. galaxies the unblemished. Thus, free they streaming last two requirements remainmodelling illusive. of the The complexmassive most hydrodynamical stars, common feedback will belief processes, allow is the from that observations central and more black predictions sophisticated holeswe to and must be blend reconciled [46–48]. acame combination from of [12] MOND whoto and produce added DM the MOND together. measured andMOND acoustic clusters An 2 peaks eV of early galaxies in active attempt [13, MOND the neutrinos]49 of CMB, with together. due this nor an to type satisfy This phase(under 11 the eV space the was ansatz requirement constraints. sterile not that of [50] neutrino, MOND able DM suggested is combining in which irrelevant at was shown to be consistent with the CMB JCAP10(2014)079 = z (2.2) (2.1) , from the ordinary matter a , ] /a, N ) ¯ ρ Φ ∇ − ) ρ y ( ( , at scale factor ν [ N πG – 3 – ∇ · = 4 ) function is parametrised as per eqs. (51) and (53) ν N Φ = Φ 2 2 to form until then and galaxies are clearly formed long ∇ ∇ being the MOND acceleration constant chosen here to be o begin a , with a . The interpolating ( o 1 − /a N pc 2 Φ ) 1 ∇ that includes baryons and neutrinos. This would also include cold dark matter − ρ = The simple options to blend MOND with DM are therefore ruled out. There areIn the also traditional framework of MOND, there exists no dark matter in galaxies. There- In this article we run MOND cosmological simulations in a framework where the modi- y 6 ( km s . and 3 density if there was anypotential in by solving our model. The QUMOND potential, Φ, is found from the Newtonian is solved to give the Newtonian potential, Φ 2 Background and method The Quasi-Linear formulation ofversion of MOND the (QUMOND; Poisson [68])simulations equation. requires Specifically, solution the ordinary of Poisson a equation modified for cosmological more involved ideas, such as theMOND tantalising theory, ([64 of dipolar65 ]), dark which matterstill, [62 require exist63] with and further bimetric sterile investigation. neutrinos and However, MOND. a simplerfore, framework it may isextended assumed frameworks, where that MOND only isbeen blended the assumed with that some baryons species thefield. produce of baryons On neutrinos, a and the it modified other the hasMOND) hand, always neutrinos where gravitational there the contribute field. do dark to exist matternot the modified has participate In a modified theories (e.g. the mutual of [66, gravitational fifth-forcemodified gravity67]). interaction, (completely gravitational to Here different field, which we to but the gothe the baryons the benefits do sterile other neutrinos of way, and do addingof suggest not. sterile the the This neutrinos CMB baryons can to produce andthe in MOND a solving principle problems — the retain engendered such missing all as when— mass addressing like the problem the overproducing sterile in acoustic massive neutrinos MONDtrino peaks clusters. clusters. produce framework, the a Furthermore, It effectiveness in modifiedconstant also of the of might gravitational MOND original MOND evade field had MOND hadway. to + to This be be sterile was “switched substantially to neu- off”a avoid decreased) model MOND (i.e. prior where altering the to the the acceleration recombinationnot sterile shape in neutrinos be of an do necessary. the not ad CMB produce hoc acoustic a power modified spectrum. gravitational field,fied In this gravitational would field is producedthe only framework by and the describe baryons,lations not the in the simulation section3 sterile setup neutrinos.and in in We section2, present section4 showwe the give results our of conclusions. our simu- aid of a dark(if it matter is halo possible (cold,gravitational at attraction warm all) between or is the hot)1, only baryons. and then possible galaxy galaxies Thus, with formationbefore if would the this. without MOND not added dark was benefit matter not of in stronger effect than until Newtonian JCAP10(2014)079 (2.4) (2.3) = 0. z -function. ν , /α 1 /a ) # s /a. 2 ν ) ¯ / b ρ 1 ¯ ρ ) − α = 2 is the standard − − s 19.6) are used to solve the Poisson equation ν b y α § ρ ρ 2 ( ( πG πG – 4 – = 4 = 4 1 + (1 + 4 s " N,b N,ν Φ -function and Φ 2 ) = . ν 2 y ∇ ( 10 ∇ α − ν = 1 is the so-called simple The procedure repeats from the second stage until the simulation reaches Interpolate to each particle’sparticle position with a to second find order the leapfrog. appropriate gravity and move each The Poisson solving step isQUMOND repeated, potential, this Φ, time which with we the take new the source gradient density of to to give find the the gravity at each cell. Multi-grid methods (see Numerical Recipes The divergence of thethe vector in source the of square the brackets QUMOND of potential. eq. (2.2) is taken which gives us Particle positions and velocitiesto are the read various in cells and with the the density cubic of the cloud-in-cell particles method. is assigned to find the Newtonianwith potential the (eq. new (2.1)). approximationfractional A of accuracy 3D the of black-red potential 10 sweep in to that update cell the and cells we iterate until we have The specifics of how to solve eqs. (2.1)&(The2.2) code are written explained for in AD11 AD11 and and described further tests therein is particle-mesh based, with a grid- Our one-component simulations are described by the following procedure: α • • • • • • 2.2 Modified BaryonicThe Dynamics modification ofsterile gravity neutrinos we and propose baryons are is found the by following: solving the the Newtonian Poisson Newtonian equation potentials of the of the code were presented in A13. mesh that hasparticles a per limit dimension. of Particle-meshPoisson solvers 257 equation. are cells required inparticle to Direct handle each or gravities. the dimension. tree-code non-linearMOND MOND The methods equation In full fail of all spherical because MOND symmetry our in Poisson does simulations not MOND equation2.1 we satisfy we must use the cannot conservation be 256 One-component laws. co-add solved simulations Our because original the one-component simulations trivial less from particle AD11 to andcommon A13 represent in use the “DM a particle only” singlemodification N-body phase-space to type simulations. the density of The original of collision- code simulationscollisionless both to particle used represents allow in DM the two-component this DM, simulations. and and articlebaryons. That another use baryons, Hereafter is, (still we a one collisionless) as describe small particle species the is representssimulations difference of the between and the how one-component this and two-component facilitatesgravitational our potentials ability from to the test DM our and hypothesis from by the generating baryons. different of [2] where and JCAP10(2014)079 . s N,ν (2.5) (2.6) (2.7) Φ COSMICS/ −∇ ) on the grid and s ν ρ should be replaced i package of [72] which , where Ω GRAFICS-2

, . ] M . | 3 b s ) N,b Φ p Φ N,ν ∇ , as per eq. (2.4)) using finite differencing s /N Φ ∇ + ) ∇ y a s N,ν box ( o + L ν a [ ( N,ν i Φ – 5 – N,b Ω ∇ · ∇ Φ 11 = |∇ 10 . Therefore, the stronger the gravitational field of the b Φ = ν Φ × = 2 ∇ 4 y . ∇ = 1 m for each species. The masses of the sterile neutrino and baryonic 3 . It is then perfectly straight-forward to code the above expressions b or Ω s ν package of [70] to generate our initial conditions. We chose to input our own transfer Read in the positions of the sterile neutrinos, compute their density ( then solve for their Newtonian potentialand (Φ multi-grid methods as described in AD11. Note that, at this point, it is a phenomenological approach at the classical level: whether In order to exploit the MBD formulation we use the The total modulus of the gravitational field experienced by any particle, sterile neutrino From eq. (2.5) we see that the unmodified (Newtonian) gravitational field of the sterile • (eqs. (2.4)–(2.7)). 2.2.2 Two-component simulations We then proceed as follows: with either Ω generates two setsone of chooses), particles: giving 256 baryons and sterile neutrinos (or any other DM particle particles are found from a Lagrangian producing thesecovariant level) equations could of be motion theLagrangian can will subject require be of couplings of further foundthereby the work, (both baryons leading if and to at necessary. sterile a the neutrinos It violation tointeresting classical of is two consequences different the and plausible metrics, in weak-equivalence that principle, their such which own a would right have (see, additional 2.2.1 e.g., [69]). Initial conditionsIn and our simulation one-component setup simulations (AD11 and A13) we made use of the original where crucially sterile neutrinos (or thegravitational baryons field. independently) the However, weaker the theSo gravitational modification all field particles to of experience the the an baryonic sterileile unmodified neutrinos neutrinos, Newtonian is gravitational as never field well modified. generatedvery as by modification the a is ster- modified influenced by gravitational the field Newtonian generated gravitational by field the of baryons, the where sterile this neutrinos. GRAFICS functions using the massivethe neutrino parametrisation resulting of linear [71]A13 matter (their figure power equations 1. (10)–(12)) spectra and were plotted for a sample of neutrino masses in or baryon, is neutrinos impacts the modificationclusion in of the the interpolating gravitational function field of the baryons, due to its in- The gravitational field produced by the sterile neutrinos is not modified and is thus Given that a modified gravitationalthe field baryons, (relative we to refer Newtonian to gravity) this is only model produced as by Modified Baryonic Dynamics (MBD). On the other hand, the gravitationalof field the produced QUMOND by the equation baryons (eq. is found (2.2)), from which the is analogue JCAP10(2014)079 ν M ) = r ( m and M b = 0. ) and then = 0. b z M ρ z at 8 σ typically is for CDM , is used to normalise 8 σ rms-PS Q = 1. This, in effect, gives us ν , by solving eq. (2.5) using finite b , as a free parameter to fit the amplitude )=(0.049, 0.267, 0.683, 0.671, 0.962). s n , rms-PS h – 6 – , term only operates on the baryonic mass profile Q Λ ν ,Ω s , as per eq. (2.4)). ν N,b ,Ω b ], the QUMOND source density for the baryons, which uses ), where the r N,b ( ν Φ ∇ M ) y + ( ν [ ) r ( b o ∇ · a + 2 ν r GM ν The procedure repeats from the second stage until the simulation reaches Interpolate to each particle’sparticle position with a to second find order the leapfrog. appropriate gravity and move each Add the QUMOND potentialfor for the the sterile baryonsgravitational neutrinos together field (as with is in derived the to eq. Newtonian move (2.7)) potential both to sets of give particles. the total potential, from which the differencing. Find the QUMOND potential for the baryons Φ Read in the positions of the baryons, compute their density on the grid ( eq. (2.6) as an argument for the interpolating function. Derive their Newtonian potential (Φ The clear difference between the one-component simulations and the two-component For the initial conditions, we employ the new CMB results provided by the PLANCK We use the quadrupole temperature, ) r • • • • • • ( b since the modification to gravity only affects the baryonic gravitational field. simulations is thattional in field. the As latter perwith the A13, a DM it is theoretical generates notsimulations expectation a possible where Newtonian because to we compare (not none follow our MONDian) exists. two-component all gravita- MBD the simulations We above can steps however run but two-component set mission ([20]) which identifies (Ω a Newtonian simulation producedthe by one-component the Newtonian MBDquite simulations well. machinery. reproduced The In thecomponent two-component A13 theoretical simulations, Newtonian we as halo simulations expected. demonstrated are massmass that Unfortunately, indistinguishable function function there from to is the no compareMOND MOND one- works with, or very but MBD well. as theoretical can be seen in figure 1 of [5], theof gravity the solver cluster in massthe function initial in power MBD.simulations, spectrum The because CMB of one quadrupole, cannot perturbations use in linear the theory in same MOND3 way to as estimate Results 3.1 Newtonian equivalentUsing mass two-component simulations with sterile neutrinos,sterile the neutrino baryons distribution, almost exactly whichMOND trace is the simulations not (A13), surprising. when computingwe As the use with mass function the the to original Newtonianhalos compare single equivalent with would component observations, mass appear of toa halos have case — if that of probedsterile is separately using neutrinos the running Newtonian dynamical (which dynamics. the massfor gives AMIGA With our our identical halo MBD MOND cluster halo it finder sized numbersM is ([73, halos). and merely ])74 halo Then on particle we both mass calculate distributions the the baryons Newtonian and equivalent mass, JCAP10(2014)079 3. . ) = 14

2 to make . /M 200 2 and extend to . ) = 14 M (

10 /M ) = 14

200 M ( /M 10 200 M ( 10 , but three neutrino masses 50, 150 and µK is required to have a mass function amplitude 5 . – 7 – µK = 4 5 = 0 regardless of initial normalisation. The mass < z ) encloses an average density of matter 200 times the r rms-PS ( Q m rms-PS M Q (black, blue, red and green shaded curves respectively). Each µK ), in pure MOND showed no variation (figure 7, A13) i.e. the mass rms-PS Q = 4, 5, 9 and 17 In figure2 we plot the mass functions from simulations with the same normalisation In general, the mass functions found using different normalisations, through the CMB From figure1 we see that In figure1 we plot the mass functions from simulations with the same neutrino mass The limited spatial resolution allows power on smaller scales to be leaked to larger rms-PS through the CMB quadrupole, of functions found with MBD clearly vary with normalisation. quadrupole ( functions had the same amplitude at which is comparable to the data (for a 100 eV sterile neutrino). the right hand sideQ of the plot.shaded The curve four demonstrates normalisations thesponds used, Poisson to in error. the increasing computed I.e.to amplitude, mass the the are function top computed plus side massfunctions the of with function Poisson data each minus error. points shaded thevirial from curve The Poisson [21] theorem corre- under (circles) error. and and side the [22] We corresponds caustic (squares compare technique and respectively). our triangles simulated found mass using the The 256 Mpc/h box simulations begin at roughly log of 100 eV, butsimulations various extend normalisations from through the left the hand CMB side quadrupole. of the The plot until 128 roughly Mpc/h log box the renormalisation, butwas there used. would The befunction renormalisation very amplitude, predominantly and little results one in differenceThis could an makes if equally increase little, an well if of use any, average 0.4 difference this dex of to fixed in many the value the conclusions bins for mass drawn the in renormalisation. this article. scales. This cansimulations be with seen Newtonian gravity clearly and compares inthe them 64 figure with Mpc/h a 2 and theoretical of 128 massof function. A13 Mpc/h the boxes, which For theoretical the plots mass simulationsand the generally function, over-predict capture mass except the for function they correct larger using under-predict amplitude of masses our for the (larger lower theoretical halos). masses massuse (smaller function For the halos) was the 256 generally 256 Mpc/h under-predictedreasonable Mpc/h boxes for to boxes, (and all thus increase the masses. probeof amplitude the Therefore, out the amplitude to to 128 of slightly larger their Mpc/h halo mass boxes. masses), functions we felt to We it use agree was with a the single amplitude mass bin at log represent the particle masswe profiles must of find baryons the andcritical radius sterile density neutrinos where and respectively. build the From mass here function. 3.2 Renormalising theWe mass ran 12 function two-component simulationsquadrupole with different and initial normalisations different throughsimulations: sterile the CMB neutrino one with masses. abecause we 128 For do Mpc/h each not have box set adaptivea mesh and of larger refinement the built range parameters, into other of our wespatial with code, masses resolution, ran and a in they it two 256 the have allows a us Mpc/h halothe limited to mass box. halo cover box, function. mass the We range Although did more meaningwith the poorer this massive the statistic smaller the i.e. larger boxes most the box, havethe larger massive higher mass we halo function. have it insufficient can spatial form. resolution However, which in leads the to simulations a suppression of JCAP10(2014)079 and = 2. α µK = 1 and α (black, blue, red and 5 and 12.5 . µK = 4 rms-PS papers). 3. The 256 Mpc/h box simulations Q . function), where the other MBD ν ) = 14 = 4, 5, 9 and 17 CDM

Λ function (eq. (2.3)) employed, we ran two /M in ν 8 rms-PS 200 σ Q M ( – 8 – 10 function). We chose 2 and extend to the right hand side of the plot. The four . ν = 2 (the standard function in MBD will still require a low normalisation for ν α ) = 14

2 CMB acoustic power spectrum and the one fitted to the mass /M 200 l M ( = 1 (the simple 10 α . Plot of the mass functions from simulations with the same neutrino mass of 100 eV, but To test if there was any sensitivity to the begin at roughly log Figure 1 various normalisations through thethe CMB left quadrupole. hand side The of 128 the Mpc/h plot box untilnormalisations simulations roughly used, extend log in from increasing amplitude, are green shaded curves respectively).mass function The plus topminus the side the of Poisson Poisson each error. error. shadedand Rines, curve The Diaferio The corresponds under & data to sidecaustic Natarajan points the technique (2008) corresponds respectively). come computed (blue to and from the red Reiprich squares computed & found mass (2002) using Böhringer the function (black virial squares) theorem and300 the eV. We find thatfunction the lower and the better sterile the neutrinosterile mass, agreement neutrino the with masses lower the between the observed 50allow amplitude a cluster and of fit 300 mass the eV, to mass function. the butis cluster using We mass always a only function. required. mass tested However, of regardless 30 of eV mass, or a lower very would lowfurther not normalisation MBD simulationssimulations using used the result was thatThis there means is using very any little reasonable the difference initial between spectrum simulations of with perturbations, through the CMB3.3 quadrupole. The significance ofThere are the three CMB values quadrupole ofthe the one CMB fitted quadrupole mentioned tofunction in here the this (more article. often The used measured in one, the form of JCAP10(2014)079 µK 5 . 3. The . 4 ≈ = 2 coeffi- l ) = 14

rms-PS /M Q 200 M ( 2 matches the observed 10 ([76]) is required so that the l 2 and extend to the right hand . µK 5 ). This leads to . 2 ) = 14 ), but diﬀerent sterile neutrino masses

µK µK /M 5 . 200 = 4 M ( should be the same for each probe. The fitted 10 – 9 – 8 σ rms-PS Q .[75] and [76] plot likelihood versus quadrupole value 2 / 1 at maximum likelihood. TT π 2 4 C 5 µK , which is roughly one sigma from the observed quadrupole. = µK 5 < model, a fitted quadrupole of around 21 rms-PS (the maximum likelihood is around 160 Q 2 CDM rms-PS Λ Q , as µK . Plot showing the mass functions from Modified Baryonic Dynamics simulations with TT 2 50 C The fitted CMB quadrupole MBD requires to be consistent withIn the the galaxy cluster mass The measured CMB quadrupole value is related to the multipole moment ≈ TT 2 amplitude of the theoretical CMBspectrum. power There spectrum should at be multipoles consistencyCMB between and the value other of cosmological theand quadrupole probes required such the to as fit galaxy baryonic the cluster acoustic mass oscillations, function galaxy clustering i.e. cient, in their figures 3towards and lower values 38 in respectively. theC latest The paper allowed and range currently theat at lower one the one sigma one sigma and sigma confidence roughly level level 8 is has shifted function is Figure 2 the same CMB quadrupole(starting from normalisation the ( bottombox simulations shaded extend curve): from the256 50 left Mpc/h (black), hand 150 box side (blue) simulations ofside the and begin of plot 300 the at eV until plot. roughly roughlythe (red). log log The Poisson The top error. 128 sideerror. Mpc/h of The each The under shaded data curve& side points corresponds Natarajan corresponds come to (2008) from to the (bluerespectively). Reiprich computed the and & mass computed red Böhringer(2002) function (black squares mass plus squares) found function and using minus Rines, the the virial Diaferio Poisson theorem and the caustic technique JCAP10(2014)079 and the µK 5 , but is between < 8 σ rms-PS Q ). This is crucial because we µK was required to have an adequate 4 . µK (21 5 lower than the measured value from CMB < σ CDM Λ rms-PS – 10 – Q case. , so they are hugely discrepant. This is far more serious CDM fits to the CMB as a basis for any MOND plus sterile neutrino Λ µK 4 . value) and found from matching the galaxy cluster mass function 21 CDM ≈ Λ CDM . Λ rms-PS Q rms-PS Q as per the µK The CMB quadrupole required is around 1 We conclude therefore that the simple model considered here is not viable, and that For this reason, we proposed that the modified gravitational field of MOND is only pro- There are similarities between our MBD model and many of the fifth force models refer- Our MBD model is far more conducive to producing a halo mass function that satisfies It was found that a normalisation of In the MBD case, we also expect the fitted quadrupole used to normalise the CMB 4 . match and a lower sterile neutrino mass wasobservations preferred ([76]). over a higher Moreto one. significantly, fit this the is CMBpreviously much adopted acoustic lower the than power the spectrumtheory, under normalisation in the required assumption the two models would givemore an involved identical MOND CMB models powerderived spectrum. should from a be fundamental modification considered ofphenomenon if the at Lagrangian the galaxy of MOND scales. Nature rather phenomenology These than is theories an emergent must indeed include at least one new degree of freedom CMB requires duced by the baryons.field. In this However, setup, the the baryonsthe sterile — neutrinos sterile subject produce neutrinos to a and the Newtonian baryons) total gravitational — Newtonian produce gravitational field aenced (both MOND-like in gravitational from [77]. field. For instancematter [67] (not employed by a the long baryons),ing range to the scalar increase colliding field, the bullet generated cluster. mutual onlyNewtonian acceleration by In gravitational of the our attraction, dark the MBD whilst two model, clustersgravitational the only compris- field. the sterile baryons neutrinos generate produce a a stronger than purely Newtonian observational constraints than ourthe previous normalisation attempts. ofquadrupole, Nevertheless, the it primordial requires power fine spectrum, tuning as is often done, through the CMB than a single, consistentthe measured fitted one quadrupole — (to as the in the CMB and other4 probes) overestimating Conclusion In previous works (AD11sterile neutrinos and could A13) produce weit the tested could observed not. the mass function This hypothesisdynamics meant of that that, on clusters combining if of galaxy MOND MOND galaxies.are is scales, with We the not then found correct as description eithercovariant of described the weak-field MOND gravitational above whole theories), (see, cosmologypower and and/or e.g., spectrum yet the (a section would highly initial 9.2neutrinos conspire contrived conditions in of scenario), to the or [2] produce same perhaps for way the MOND as a does correct the discussion not baryons. CMB affect in angular the the sterile context of to be the same. Clearly, the galaxy cluster mass function requires (21 quadrupole to the CMBone is and generally consistent two with sigmaone most larger other can than probes the be of measuredalternative achieved one. gravity by theories, However, models consistencymodelling). etc) that with or suppress the increase the measured the fitted measured one quadrupole (due (due to to inflationary foreground theories, JCAP10(2014)079 ]. ]. ] ] , ] 527 SPIRE SPIRE (2013) IN IN forest [ ][ α 554 Mon. Not. , Mon. Not. Roy. , arXiv:0906.3322 N-body simulations arXiv:1112.3960 eV sterile neutrinos (2008) 1470 arXiv:1006.1647 11 arXiv:1310.6148 Astron. Astrophys. , , 387 arXiv:1401.1146 arXiv:1303.0995 (2014) 746 , (2012) 10[ ]. (2010) 395[ ? Astron. Astrophys. 15 440 , ]. (2010) A32[ 402 ]. CDM SPIRE Λ IN ]. 523 ][ SPIRE (2012) A76[ SPIRE The dynamics of the bulge dominated IN IN ][ SPIRE 543 ][ IN THINGS about MOND Living Rev. 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